New approach to solutions of a class of singular fractional q-differential problem via quantum calculus

Abstract

In the present article, by using the fixed point technique and the Arzelà–Ascoli theorem on cones, we wish to investigate the existence of solutions for a non-linear problems regular and singular fractional q-differential equation

$$ \bigl({}^{c}D_{q}^{\alpha }f\bigr) (t) = w \bigl(t, f(t), f'(t), \bigl({}^{c}D_{q}^{ \beta }f \bigr) (t) \bigr), $$

under the conditions $f(0) = c_{1} f(1)$, $f'(0)= c_{2} ({}^{c}D_{q} ^{\beta } f) (1)$ and $f''(0) = f'''(0) = \cdots =f^{(n-1)}(0) = 0$, where $\alpha \in (n-1, n)$ with $n\geq 3$, $\beta , q \in J=(0,1)$, $c_{1} \in J$, $c_{2} \in (0, \varGamma _{q} (2- \beta ))$, the function w is $L^{\kappa }$-Carathéodory, $w(t, x_{1}, x_{2}, x_{3})$ and may be singular and ${}^{c}D_{q}^{\alpha }$ the fractional Caputo type q-derivative. Of course, here we applied the definitions of the fractional q-derivative of Riemann–Liouville and Caputo type by presenting some examples with tables and algorithms; we will illustrate our results, too.

1 Introduction

The fractional calculus and q-calculus deal with the generalization of integration and differentiation of integer order to any order. It is known that fractional calculus is used for a better description of phenomena having both discrete and continuous behaviors, and applying in different sciences and engineering such as mechanics, electricity, biology, control theory, signal and image processing [1–12]. It has an old history and several fractional derivations where defined, such as the Caputo, the Riemann–Liouville and the Caputo and Fabrizio derivations. These derivations appeared recently in much work on integro-differential equations by using different views which young researchers could use for their work [13–27]. The fractional q-calculus has been applied to almost very field of non-linear mathematics analysis [28–38]. This branch of mathematics was introduced by Jackson in 1910 [1, 39]. For earlier work on the topic, we refer to [40, 41], whereas the preliminary concepts on q-fractional calculus can be found in [4]. For some applications of the q-fractional calculus, see for example [2, 3, 5, 7, 8, 42–44]. Also, there has been a significant increase in knowledge in the field of differential and q-differential equations and inclusions in recent years [45–49].

In 2012, Ahmad et al., studied the existence and uniqueness of solutions for the fractional q-difference equations ${}^{c}D_{q} ^{\alpha }u(t)= T ( t, u(t) )$ with the boundary conditions $\alpha _{1} u(0) - \beta _{1} D_{q} u(0) = \gamma _{1} u(\eta _{1})$ and $\alpha _{2} u(1) - \beta _{2} D_{q} u(1) = \gamma _{2} u(\eta _{2})$, where $\alpha \in (1, 2]$, $\alpha _{i}$, $\beta _{i}$, $\gamma _{i}$, $\eta _{i}$ are real numbers, for $i=1,2$ and $T \in C(J \times \mathbb{R}, \mathbb{R})$ [34]. In 2013, Baleanu et al., reviewed the nonlinear singular fractional problem $({}^{c}D^{\alpha }u)(t) = w (t, u(t), u'(t), ({}^{c}D ^{\beta }u) (t) )$, under the boundary conditions $u(0) = a_{1} u(1)$, $u'(0)= a_{2} ({}^{c}D^{\beta } u) (1)$ and $u''(0) = u'''(0) = \cdots = u^{(n-1)}(0) =0$ on cones, where $\alpha \in (n-1, n)$ with an integer number $n \geq 3$, β, $a_{1}$, $a_{2} \in J=(0,1)$, $(-\infty , 1)$, $(0, \varGamma (2- \beta ))$, respectively, and w is a $L^{\kappa }$-Carathéodory function, $\kappa (\alpha -1) > 1$, with the same conditions, which is was addressed by Agarwal et al. [50]. In 2013, Zhao el al. [38] reviewed the q-integral problem $(D_{q}^{\alpha }u)(t) + f(t, u(t) )=0$, with the conditions $u(1)$, $u(0)$ equal to $\mu I_{q}^{\beta }u(\eta )$, 0, respectively, for almost all $t \in (0,1)$, where $q \in (0,1)$ and α, β, η belong to $(1, 2]$, $(0, 2]$, $(0,1)$, respectively, μ is positive real number, $D_{q}^{\alpha }$ is the q-derivative of Riemann–Liouville and real-values continuous map u defined on $I \times [0, \infty )$. In 2014, Jiang et al., investigated the existence and uniqueness of solution of the problem $D_{q}^{\beta }( \phi _{p}( D_{q}^{\alpha }y(x)))+ w ( x, y(x), D_{q}^{\gamma }y(x) )=0$, under the conditions $y(0) =D_{q} y(0)= D_{q}^{\alpha }y(0)=0$ and $y(1) = \mu I_{q} y(\eta )$, by invoking the p-Laplacian operator, where w belongs to $C(E, \mathbb{R} )$ with $E=[0,1] \times \mathbb{R}^{2}$, α and β, q, η, γ belong to in $(2,3)$ and $(0,1)$, respectively, $\mu >0$ is constant, $D_{q}^{\alpha }$ is the fractional q-derivative of the Riemann–Liouville type, $D_{q}$ and $I_{q}$ denote the q-derivative and the q-integral, receptively, and $\phi _{p}$ is the p-Laplacian operator defined by $\phi _{p} (s) = |s|^{p-2} s$, with $p>1$ [51].

Two year later, in 2016, Abdeljawad et al. [52] stated and proved a new discrete q-fractional version of the Gronwall inequality: $({}_{q}C_{a}^{ \alpha }f)(t) = w( t, f(t) )$ and $f(a) =\gamma$ such that $\alpha \in (0, 1]$, $a \in \mathbb{T}_{q}= \{q^{n}: n \in \mathbb{Z} \}$, t belongs to $\mathbb{T}_{a}= [0, \infty )_{q} = \{ q^{-i} a: i=0, 1, 2, \ldots \}$, ${}_{q}C_{a}^{\alpha }$ means the Caputo fractional difference of order α and $w(t, x)$ fulfills a Lipschitz condition for all t and x. Then, in 2017, Zhou et al. [53] provided existence criteria for the solutions of the fractional Langevin differential equation under some conditions:

$$ \textstyle\begin{cases} D_{0^{+}}^{\beta }\phi _{p} [ ( D_{0^{+}}^{\alpha }+ \eta ) f(t) ] = w(t, f(t), D_{0^{+}}^{\alpha }f(t)), \\ f(0) = -f(1), \qquad D_{0^{+}}^{\alpha }f(0) = - D_{0^{+}}^{\alpha }f(1), \end{cases} $$

and

$$ \textstyle\begin{cases} {}_{q}D_{0^{+}}^{\beta }\phi _{p} [ ( D_{0^{+}}^{\alpha }+ \eta ) f(t) ] = w (t, f(t), {}_{q}D_{0^{+}}^{\alpha }f(t) ), \\ f(0) = -f(1), \qquad {}_{q}D_{0^{+}}^{\alpha }f(0) = - {}_{q}D_{0^{+}}^{\alpha }f(1), \end{cases} $$

for each $t \in [0,1]$, where $0 < \alpha , \beta \leq 1$, η is larger than or equal to zero, $1 < \alpha + \beta <2$, $q \in (0,1)$, and $\phi _{p} (s) = |s|^{p-2} s$, with $p \in (1, 2]$. In 2017, Baleanu et al., presented a new method to investigate some fractional integro-differential equations involving the Caputo–Fabrizio derivation,

$$ {}^{\mathrm{CF}}D^{\alpha }u(t) = \frac{(2-\alpha ) M(\alpha )}{2 (1-\alpha )} \int _{0}^{t} \exp \biggl( \frac{\alpha }{\alpha -1} (t-s) \biggr) u'(s) \, ds, $$

where t is used and $M(\alpha )$ is a normalization constant depending on α such that $M(0) =M(1) = 1$; one proved the existence of approximate solutions for these problems [16]. In the same year, they introduced a new operator entitled the infinite coefficient-symmetric Caputo–Fabrizio fractional derivative and applied it to the investigation of the approximate solutions for two infinite coefficient-symmetric Caputo–Fabrizio fractional integro-differential problems [17].

In addition to, Akbari et al., by using the shifted Legendre and Chebyshev polynomials, discussed the existence of solutions for a sum-type fractional integro-differential problem under the Caputo differentiation [19]. Over the past three years, Baleanu and Rezapour et al., by using the Caputo–Fabrizio derivative, achieved innovation, and remarkable and interesting results were found for solutions of fractional differential equations [13–16, 18, 20–25]. In the next year, Rezapour et al., investigated the existence of solutions for the inclusion ${}^{c}D^{\alpha }x(t) \in F (x, f(x), {}^{c}D^{\beta }f(x), f' (x) )$ for each $x\in I$ with the conditions ${}^{c}D^{\beta } f(0) -\int _{0}^{\eta _{1}} f(r) \,dr= f(0) + f' (0)$ and ${}^{c}D^{\beta } f(1) - \int _{0}^{\eta _{2}} f(r) \,dr = f(1) + f' (1)$, where the multifunction F maps $[0,1] \times \mathbb{R} ^{3}$ to $2^{ \mathbb{R}}$ and is compact valued and ${}^{c}D^{ \alpha }$ is the Caputo differential operator [54].

In 2019, Samei et al., discussed the fractional hybrid q-differential inclusions ${}^{c}D_{q}^{\alpha } ( x / F ( t, x, I_{q}^{\alpha _{1}} x, \ldots , I_{q}^{\alpha _{n}} x ) ) \in T ( t, x, I_{q}^{\beta _{1}} x, \ldots , I_{q}^{ \beta _{k}} x )$, with the boundary conditions $x(0) =x_{0}$ and $x(1)=x_{1}$, where $1 < \alpha \leq 2$, $q \in (0,1)$, $x_{0}, x _{1} \in \mathbb{R}$, $\alpha _{i} >0$, for $i=1, 2, \ldots , n$, $\beta _{j} > 0$, for $j=1, 2, \ldots , k$, $n, k\in \mathbb{N}$, ${}^{c}D_{q}^{\alpha }$ denotes a Caputo type q-derivative of order α, $I_{q}^{\beta }$ denotes the Riemann–Liouville type q-integral of order β, $F: J \times \mathbb{R}^{n} \to (0, \infty )$ is continuous and T mapping $J\times \mathbb{R}^{k}$ to $P (\mathbb{R})$ is a multifunction [32]. Also, they discussed the existence of solutions for the fractional q-derivative inclusions ${}^{c}D_{q}^{\alpha }x(t) \in F ( t, x(t), x'(t), {}^{c}D_{q}^{\beta }x(t) )$, $x(0) + x'(0) + {}^{c}D_{q}^{ \beta }x(0) = \int _{0}^{\eta _{1}} x(s) \, ds$, and $x(1) + x'(1) + {}^{c}D_{q}^{\beta }x(1) = \int _{0}^{\eta _{2}} x(s) \, ds$, for any t in I and $q, \eta _{1}, \eta _{2}, \beta \in (0,1)$, where F maps $I\times \mathbb{R}^{3}$ into $2^{\mathbb{R}}$ is a compact valued multifunction and ${}^{c}D_{q}^{\alpha }$ is the fractional Caputo type q-derivative operator of order $\alpha \in (1, 2]$, and $\varGamma _{q} (2- \beta )(\eta ^{2} \nu - \nu ^{2} \eta - \eta ^{2} + \nu ^{2} + 4 \eta - 2\nu -2) + 2(1-\eta ) \neq 0$, such that $\alpha -\beta >1$ [49]. In 2019, Samei et al. [32, 36], investigated the fractional hybrid q-difference inclusion, and also equations and inclusions of multi-term fractional q-integro-differential equations with non-separated and initial boundary conditions.

In this article, motivated by the main idea of the literature, we are going to investigate the problems of the fractional q-differential equation

$$ \textstyle\begin{cases} ({}^{c}D_{q}^{\alpha }f)(t) = w (t, f(t), f'(t), ({}^{c}D_{q}^{ \beta }f)(t) ), \\ f(0) = c_{1} f(1), \\ f'(0)= c_{2} ({}^{c}D_{q}^{\beta } f) (1), \\ f''(0) = f'''(0) =\cdots = f^{(n-1)}(0) = 0, \end{cases} $$

(1)

where $\alpha \in (n-1, n)$ with $n\geq 3$, $\beta , q \in J=(0,1)$, $c_{1} \in J$, $c_{2} \in (0, B)$ with $B=\varGamma _{q} (2- \beta )$, the function w is $L^{\kappa }$-Carathéodory being positive real valued and $\kappa (\alpha -1)> 1$, $w(t, x_{1}, x_{2}, x_{3})$ may be singular at the value 0 of its space variables $x_{1}$, $x_{2}$, $x_{3}$; ${}^{c}D_{q}^{\alpha }$ is the fractional Caputo type q-derivative.

This manuscript is organized as follows: In Sect. 2, we recall some preliminary concepts and fundamental results of q-calculus. Section 3 is devoted to the main results, while examples illustrating the obtained results and algorithm for the problems are presented in Sect. 4.

2 Preliminaries

First of all, we summarize the basic definitions and properties of q-calculus and q-fractional integrals and derivatives. One can find more information about them in [1–6, 8].

Suppose that $q \in (0,1)$ and $a \in \mathbb{R}$. Define $[a]_{q}=\frac{1-q ^{a}}{1-q}$ [1]. The power function $(x-y)_{q}^{n}$ with $n \in \mathbb{N}_{0}$ is $(x-y)_{q}^{(n)}= \prod_{k=0}^{n-1} (x - yq^{k})$ and $(x-y)_{q}^{(0)}=1$ where $x, y \in \mathbb{R}$ and $\mathbb{N}_{0} := \{ 0\} \cup \mathbb{N}$ [1–3]. Also, for $\alpha \in \mathbb{R}$ and $a \neq 0$, we have $(x-y)_{q}^{(\alpha )}= x^{ \alpha }\prod_{k=0}^{\infty }(x-yq^{k}) / (x - yq^{\alpha + k})$. If $y=0$, then it is clear that $x^{(\alpha )}= x^{\alpha }$ (Algorithm 1). The q-Gamma function is given by $\varGamma _{q}(z) = (1-q)^{(z-1)} / (1-q)^{z -1}$, where $z \in \mathbb{R} \backslash \{0, -1, -2, \ldots \}$ [1, 2, 55, 56]. Note that $\varGamma _{q} (z+1) = [z]_{q} \varGamma _{q} (z)$. We show in Algorithm 2, a pseudo-code for estimating the q-Gamma function. The q-derivative of the function f, is defined by $(D_{q} f)(x) = \frac{f(x) - f(qx)}{(1- q)x}$ and $(D_{q} f)(0) = \lim_{x \to 0} (D_{q} f)(x)$ [2, 6, 57]. One can find in Algorithm 3 a pseudo-code for calculating the q-derivative of the function f. The higher-order q-derivative of a function f is defined by $(D_{q}^{n} f)(x) = D_{q}(D_{q}^{n-1} f)(x)$ for all $n \geq 1$, where $(D_{q}^{0} f)(x) = f(x)$ [57]. The q-integral of a function f defined on $[0,b]$ is defined by

$$ I_{q} f(x) = \int _{0}^{x} f(s) \,d_{q} s = x(1- q) \sum_{k=0}^{\infty } q^{k} f \bigl(x q^{k}\bigr), $$

for $x \in [0, b]$, provided that the series absolutely converges, which is shown in Algorithm 4 [57, 58]. If a is in $[0, b]$, then

$$ \int _{a}^{b} f(u) \,d_{q} u = I_{q} f(b) - I_{q} f(a) = (1-q) \sum _{k=0} ^{\infty } q^{k} \bigl[ b f\bigl(b q^{k}\bigr) - a f\bigl(a q^{k}\bigr) \bigr], $$

whenever the series exists. The operator $I_{q}^{n}$ is given by $(I_{q}^{0} h)(x) = h(x)$ and

$$ \bigl(I_{q}^{n} h\bigr) (x) = \bigl(I_{q} \bigl(I_{q}^{n-1} h\bigr)\bigr) (x), $$

for $n \geq 1$ and $g \in C([0,b])$ which is shown in Algorithm 5 [57]. It has been proved that $(D_{q} (I_{q} f))(x) = f(x)$ and $(I_{q} (D_{q} f))(x) = f(x) - f(0)$ whenever f is continuous at $x =0$ [2, 57, 58]. The fractional Riemann–Liouville type q-integral of the function f on J, of $\alpha \geq 0$ is given by $(I_{q}^{0} f)(t) = f(t)$ and

$$ \bigl(I_{q}^{\alpha }f\bigr) (t) = \frac{1}{\varGamma _{q}(\alpha )} \int _{0}^{t} (t- qs)^{(\alpha - 1)} f(s) \,d_{q}s, $$

for $t \in J$ and $\alpha >0$ [35, 55, 59]. Also, the fractional Caputo type q-derivative of the function f is given by

$$ \begin{aligned}[b] \bigl( {}^{c}D_{q}^{\alpha }f \bigr) (t) & = \bigl( I_{q}^{[\alpha ]-\alpha }\bigl( D_{q}^{[\alpha ]} f\bigr) \bigr) (t) \\ & = \frac{1}{\varGamma _{q} ([\alpha ]-\alpha )} \int _{0} ^{t} (t- qs)^{ ([\alpha ]-\alpha -1 )} \bigl( D_{q}^{[ \alpha ]} f \bigr) (s)\, d_{q}s, \end{aligned} $$

(2)

for $t \in J$ and $\alpha >0$ [35, 59]. It has been proved that $( I_{q}^{\beta } (I_{q}^{\alpha } f)) (x) = ( I_{q} ^{\alpha + \beta } f) (x)$ and $(D_{q}^{\alpha } (I_{q}^{\alpha } f) ) (x) = f(x)$, where α and β in $[0, \infty )$ [2, 35, 55, 59].

Algorithm 1

The proposed method for calculating $(a-b)_{q}^{(\alpha )}$

Algorithm 2

The proposed method for calculating $\varGamma _{q}(x)$

Algorithm 3

The proposed method for calculating $(D_{q} f)(x)$

Algorithm 4

The proposed method for calculating $(I_{q}^{\alpha}f)(x)$

Algorithm 5

The proposed method for calculating $\int _{a}^{b} f(r)\, d_{q} r$

Let $\overline{J}= [0,1]$ and A be a subset of $\mathbb{R}^{3}$. We denote the space of functions whose κth powers of modulus are integrable on J̅, endowed with norm $\|u \|_{\kappa }= ( \int _{0}^{1} |u(t)|^{\kappa }\, dt)^{1/\kappa }$ and the set of absolutely continuous functions on J̅, by $L^{\kappa }( \overline{J})$ and $AC(\overline{J})$, respectively, where $\kappa \in [1, \infty )$.

Definition 1

We say that f is multi-singular when it is singular at more than one point t. Also, a real-valued and non-continuous function f on the interval $I=[a, b]$ is said to be singular whenever f is non-constant on I, and there exists a set S of measure 0 such that for x outside of S the derivative $f'(x)$ exists and is zero, that is, the derivative of f vanishes almost everywhere.

Definition 2

A function w is called $L^{\kappa }$-Carathéodory on $\overline{J} \times A$ whenever the real-valued function $w(\cdot, x_{1}, x_{2}, x_{3})$ on J̅ is measurable for all $(x_{1}, x _{2}, x_{3} )$ belonging to A, the real-valued function $w(t, \cdot, \cdot, \cdot)$ defined on A is continuous for each t and belongs to $(0,1]$ and for each compact set $C\subset A$, there exists $\varphi _{C} \in L^{\kappa }( \overline{J})$, such that $|w(t, x_{1},x _{2},x_{3})| \leq \varphi _{C}(t)$, for t belonging to J̅ and $(x_{1},x_{2},x_{3}) \in C$.

Definition 3

A real value function f define on J̅ is called a positive solution for problem (1), whenever $f(t)$ is more than to zero, ${}^{c}D_{q}^{\alpha } f$ is a function in $L^{\kappa }( \overline{J})$ and f satisfies the boundary conditions for all $t \in \overline{J}$.

Throughout the paper, we suppose that the function w in (1) has the following conditions:

• (H1) The map w is an $L^{\kappa }$-Carathéodory on $\overline{J} \times A$, where $\kappa (\alpha -1)> 1$ and it fulfills the estimate

  $$ w (t, x_{1}, x_{2}, x_{3}) \leq g_{1}(x_{1}) + g_{2}\bigl( \vert x_{2} \vert \bigr) + g_{3}\bigl( \vert x _{3} \vert \bigr) + \gamma (t) \theta \bigl( x_{1}, \vert x_{2} \vert , \vert x_{3} \vert \bigr), $$

  for $t\in \overline{J}$ and $(x_{1}, x_{2}, x_{3})$ belonging to A, where positive valued functions $g_{1}$, $g_{2}$, $g_{3}$ in $C(\mathbb{R}^{>0})$ are decreasing, γ and θ in $L^{\kappa }(\overline{J})$ and $C( E )$ where $E=[0 , \infty ) \times [0 , \infty ) \times [0 , \infty )$, respectively, are positive, w is increasing in all its arguments and $\lim_{y\to \infty } \frac{w(y, y, y)}{y}= 0$ and $\varGamma _{q}(\alpha ) (I_{q}^{\alpha }g _{i}^{\kappa })(1)< \infty$ for $i=1,2,3$.

• (H2) For each $t \in \overline{J}$ and $(x_{1}, x_{2}, x_{3})$ belongs to A, there exists $m >0$ such that $m \leq w(t, x_{1}, x _{2}, x_{3})$.

Since we imagine that problem (1) is singular, that is, $w(t, x_{1}, x_{2}, x_{3})$ may be singular at the value zero of its space variables $x_{1}$, $x_{2}$ and $x_{3}$, we use regularization and sequential techniques for the existence of positive solutions of the problem. For this purpose, for each natural number n, define the function $w_{n}$ on $\overline{J}\times A$ by

$$ w_{n} ( t,x_{1}, x_{2}, x_{3} ) = w \bigl( t,\xi _{n} ^{+} (x_{1}), \xi _{n}^{+} (x_{2}), \xi _{n}^{+} (x_{3}) \bigr), $$

where $\xi _{n}^{+} (u) = u$, whenever $f\geq \frac{1}{n}$ and $\xi _{n}^{+} (u) =\frac{1}{n}$, whenever $u < \frac{1}{n}$.

Remark 1

Since w is $L^{\kappa }$-Carathéodory, obviously $w_{n}$ is an $L^{\kappa }$-Carathéodory function on $\overline{J} \times A$ and by assumption (H1), for each n, we get

$$\begin{aligned} w_{n} (t, x_{1}, x_{2}, x_{3}) \leq{} & g_{1} \biggl( \frac{1}{n} \biggr) + g_{2} \biggl( \frac{1}{n} \biggr) + g_{3} \biggl( \frac{1}{n} \biggr) \\ & {} + \gamma (t) \theta \bigl(1 + x_{1}, 1 + \vert x_{2} \vert , 1+ \vert x_{3} \vert \bigr) \end{aligned}$$

and $w_{n} (t, x_{1}, x_{2}, x_{3}) \leq g_{1}(x_{1}) + g_{2}(|x_{2}|) + g_{3}(|x_{3}|) + \gamma (t) \theta ( 1 + x_{1}, 1+|x_{2}|, 1+|x_{3}|)$. Also, the condition (H2) entails that there exists a natural number m such that $m\leq w_{n} (t, x_{1}, x_{2}, x_{3})$.

Lemma 4

([60])

Suppose thatτbelongs to $L^{\kappa }(\overline{J})$and $t_{1}, t_{2} \in \overline{J}$. Then

$$ \bigl\vert \varGamma _{q} (\alpha -1) \bigl(I_{q}^{\alpha -1} \tau \bigr) (t) \bigr\vert \leq \biggl( \frac{t^{d}}{d} \biggr)^{1/p} \Vert \tau \Vert _{\kappa }, $$

for almost alltbelongs toJ̅and

$$\begin{aligned} &\biggl\vert \int _{0}^{t_{2}} ( t_{2}- qs)^{ (\alpha - 2)} \tau (s) \, d _{q}s - \int _{0}^{t_{1}} (t_{1} - qs)^{(\alpha -2)} \tau (s) \, d_{q}s \biggr\vert \\ & \quad \leq \biggl( \frac{ t_{1}^{d} +(t_{2} -t_{1} )^{d} - t_{2}^{d}}{ d } \biggr)^{1/p} \Vert \tau \Vert _{\kappa }+ \biggl( \frac{(t_{2} -t_{1})^{d}}{d} \biggr)^{1/p} \Vert \tau \Vert _{\kappa }, \end{aligned}$$

whenever $t_{1} \leq t_{2}$, here $d-1=(\alpha -2) p$with $p=\frac{ \kappa -1}{\kappa }$.

3 Main results

At present, we discuss the existence of solutions of problem (1). Foremost, we prove the key result.

Lemma 5

Suppose thatvbelongs to $C(\overline{J})$. Then the boundary value problem

$$ \textstyle\begin{cases} ({}^{c}D_{q}^{\alpha }f)(t) = v(t), \\ f(0) = c_{1} f(1), \\ f'(0)= c_{2} ({}^{c}D_{q}^{\beta } f) (1), \\ f''(0) = f'''(0) =\cdots = f^{(n-1)}(0) =0, \end{cases} $$

(3)

for each $t\in J$, where $c_{1} \in (n-1, n)$with $n \geq 3$and $c_{2} \in (0, B)$with $B=\varGamma _{q} ( 2-\beta )$, is equivalent to the fractional integral equation $f(t)=\int _{0}^{1} G_{q}(t, s) v(s)\, d _{q}s$, for all $s, t \in \overline{J}$, where

$$ G_{q}(t, s) = \textstyle\begin{cases} \frac{(t -qs)^{(\alpha -1)}}{ \varGamma _{q} (\alpha )} + \frac{ c_{1} (1-qs)^{( \alpha -1)} }{(1-c_{1}) \varGamma _{q}(\alpha )} + \frac{ c_{2} B (c_{1} + t- c_{1} t) (1-qs)^{ (\alpha -\beta -1)}}{(1-a) \varGamma _{q}( \alpha - \beta )( B- c_{2})}, & s \leq t, \\ \frac{c_{1} (1 -qs)^{(\alpha -1)} }{ (1- c_{1}) \varGamma _{q}(\alpha ) }+ \frac{ c_{2} B ( c_{1} + t- c_{1} t)(1 -qs)^{(\alpha -\beta -1)}}{ (1 - c_{1}) \varGamma _{q}(\alpha - \beta )(B - c_{2})}, & t\leq s. \end{cases} $$

(4)

Proof

From $({}^{c}D_{q}^{\alpha }) f(t)=v(t)$, for all t belonging to $(0,1)$ and the boundary conditions $f''(0) = f'''(0) = f^{(n-1)} (0)=0$, we obtain

$$\begin{aligned} f(t) & = \bigl(I_{q}^{\alpha }v\bigr) (t) + f(0) + f'(0) t + \frac{f''(0)}{2!} t ^{2} +\cdots + \frac{ f^{(n-1)}(0)}{(n-1)!} t^{n-1} \\ & = \bigl(I_{q}^{\alpha }v\bigr) (t) + f(0) + f'(0) t. \end{aligned}$$

So, we obtain

$$ \bigl({}^{c}D_{q}^{\beta }f\bigr) (t) = \bigl(I_{q}^{\alpha -\beta } v\bigr) (t) + \bigl({}^{c}D _{q}^{\beta }\bigr) \bigl( f(0) + f'(0)t \bigr) = \bigl(I_{q}^{\alpha - \beta } v\bigr) (t) + \frac{1}{B}f'(0) t^{1- \beta }. $$

Therefore, $f(1) =(I_{q}^{\alpha } v)(1) + f(0) + f'(0)$, and $({}^{c}D_{q}^{\beta } f)(1) = (I_{q}^{ \alpha - \beta } v) (1) + \frac{1}{B} f'(0)$. By using the conditions of problem (3), we get $f(0)= c_{1} ( (I_{q}^{\alpha } v)(1) + f(0) + f'(0) )$ and $f'(0) = c_{2} ( (I_{q}^{\alpha - \beta } v)(1) + \frac{1}{B} f'(0) )$. Hence,

$$\begin{aligned} f (0) &= \frac{c_{1}}{(1 - c_{1}) } \bigl(I_{q}^{\alpha }v\bigr) (1) + \frac{c _{1} c_{2} B}{( 1 - c_{1})( B - c_{2})} \bigl( I_{q}^{\alpha -\beta } v\bigr) (1) \end{aligned}$$

and $f'(0) =\frac{c_{2} B}{ B - c_{2}} (I_{q}^{\alpha -\beta } v) (1)$. We simply observe that

$$\begin{aligned} f(t) ={}& \bigl(I_{q}^{\alpha }v\bigr) (t) \, d_{q}s + f(0) + f'(0) t \\ ={}& \int _{0}^{t} \biggl[ \frac{( t- qs)^{ (\alpha -1)}}{ \varGamma _{q}( \alpha )} + \frac{ c_{1} ( 1 - qs)^{(\alpha -1)}}{ (1 - c_{1}) \varGamma _{q} (\alpha ) } \\ & {} + \frac{ c_{2} B (c_{1} + t - c_{1} t )(1 - qs)^{(\alpha - \beta -1)}}{(1 - c_{1}) \varGamma _{q} (\alpha -\beta ) (B - c_{2})} \biggr] v(s)\, d_{q}s \\ & {} + \int _{t}^{1} \biggl[ \frac{ c_{1} (1- qs)^{(\alpha -1)} }{ (1- c_{1}) \varGamma _{q}( \alpha ) } \\ & {} + \frac{ c_{2} B( c_{1} + t- c_{1} t) (1-qs)^{(\alpha - \beta -1)}}{ (1- c_{1}) \varGamma _{q}( \alpha -\beta ) (B - c_{2})} \biggr] v(s) \, d_{q}s \\ ={}& \int _{0}^{1} G_{q}(t, s) v(s) \,d_{q}s. \end{aligned}$$

This completes our proof. □

For unification, we put $p = \frac{\kappa -1}{\kappa }$ with $\kappa \geq 1$, $d=(\alpha -2) p +1$,

$$ \begin{aligned}[b] \varLambda _{1} & = \frac{1}{ ( 1- c_{1}) \varGamma _{q} (\alpha ) } + \frac{ \varGamma _{q} ( \alpha -\beta ) ( B- c_{2} ) + c_{2} \varGamma _{q} (2- \beta )}{ ( 1 - c_{1}) \varGamma _{q} (\alpha -\beta ) ( \varGamma _{q} (2-\beta )- c_{2} )} \\ & = \frac{\varGamma _{q} ( \alpha -\beta ) ( B - c_{2} ) + c_{2} B\varGamma _{q} (\alpha ) }{ ( 1 - c_{1}) \varGamma _{q} (\alpha ) \varGamma _{q} (\alpha -\beta ) (B- c_{2} )} \end{aligned} $$

(5)

and

$$ \begin{aligned} \varLambda _{2} & = \frac{c_{1} c_{2} B}{(1- c_{1}) \varGamma _{q}( \alpha - \beta )( B - c_{2})}. \end{aligned} $$

(6)

Lemma 6

Theq-Green function $G_{q}(t,s)$in Lemma 5, which belongs to $C(\overline{J} \times \overline{J})$, for all $(t, s) \in \overline{J}\times \overline{J}$, satisfies the conditions:

• (i) $G_{q}(t, s) \leq \varLambda _{1} (1- qs)^{(\alpha -\beta -1)} \leq 1$,

• (ii) $G_{q}(t, s) \geq \varLambda _{2} (1 - qs)^{(\alpha -\beta -1)}$.

Proof

One can easy to check that $G_{q} (t, s) >0$ on $\overline{J}\times \overline{J}$. Then from (5) and (6), we have

$$\begin{aligned} &\frac{( t - qs)^{ (\alpha -1)}}{ \varGamma _{q} ( \alpha )} + \frac{ c _{1} (1-qs)^{(\alpha -1)} }{ ( 1- c_{1}) \varGamma _{q}( \alpha )} + \frac{ c_{2} B ( c_{1} + t - c_{1} t)(1- qs)^{(\alpha -\beta -1)}}{( 1- c _{1}) \varGamma _{q} (\alpha -\beta ) (B- c_{2})} \\ & \quad \leq \frac{(1 - qs)^{(\alpha - \beta -1)}( \varGamma _{q} (\alpha - \beta )( B) - c_{2}) + c_{2} B \varGamma _{q} (\alpha ) }{(1- c_{1}) \varGamma _{q}( \alpha )\varGamma _{q}(\alpha -\beta ) ( B - c_{2} )} \\ &\quad =\varLambda _{1} (1 - qs)^{(\alpha -\beta -1)} \end{aligned}$$

and

$$\begin{aligned} &\frac{ c_{1} (1- qs)^{(\alpha -1)} }{(1- c_{1}) \varGamma _{q} (\alpha ) } + \frac{ c_{2} B( c_{1} + t - c_{1} t)(1 - qs)^{(\alpha -\beta - 1)}}{( 1- c_{1}) \varGamma _{q} (\alpha -\beta ) ( B - c_{2} )} \\ &\quad \leq \frac{(1 - qs)^{ (\alpha -\beta -1)}( c_{1} (1 -qs)^{(-\beta )} }{(1- c_{1}) \varGamma _{q} (\alpha ) } \\ & \qquad{} + \frac{ c_{2} B(c_{1} + t - c_{1} t)}{( 1 - c_{1}) \varGamma _{q}( \alpha -\beta )( B- c_{2} )} \\ &\quad \leq \varLambda _{1} (1- qs)^{( \alpha -\beta -1)}, \end{aligned}$$

whenever $s \leq t$ and $t\leq s$, respectively, for t and s in J̅. Therefore, $G_{q}(t, s) \leq \varLambda _{1} (1-qs)^{( \alpha -\beta -1)}$, for all $(t, s)$ belonging to $\overline{J} \times \overline{J}$. Finally, it is observed that

$$\begin{aligned} &(1 - c_{1}) ( B- c_{2}) \varGamma _{q}( \alpha -\beta ) \varGamma _{q}( \alpha ) G_{q}(t, s) \\ &\quad \geq c_{2} B \varGamma _{q}(\alpha ) ( c_{1} + t - c_{1}t) ( 1-qs)^{( \alpha -\beta -1)} \\ &\quad \geq c_{1} c_{2} B \varGamma _{q}( \alpha ) (1- q s)^{(\alpha -\beta -1)}. \end{aligned}$$

Therefore, $G_{q}(t,s) \geq \varLambda _{2} (1-qs)^{(\alpha -\beta -1)}$ for all $(t, s)$ belonging to $\overline{J}\times \overline{J}$. □

Consider the Banach space $X= C^{1}(\overline{J})$ endowed with the norm $\| u \|_{\ast }= \max \{ \|u\|, \| u' \|\}$ and the cone P on X, containing all the functions u belonging to X such that $u(t)\geq 0$ and $u'(t)\geq 0$ for all t. Now, we define an operator $\varTheta _{n}$ on P by

$$ (\varTheta _{n} u) (t) = \int _{0}^{1} G_{q}(t,s) T_{n} \bigl(s, f(s), f'(s), \bigl({}^{c}D_{q}^{\beta } f\bigr) (s) \bigr) \, d_{q}s. $$

At present, we show that the operator $\varTheta _{n}$ is completely continuous [61].

Lemma 7

$\varTheta _{n}$is a completely continuous operator, whenever the $\varTheta _{n}$satisfy conditions (H1) and (H2) for all natural number sn.

Proof

Consider an element $u \in P$. Then $u \in C(\overline{J})$. Also, u and $u'$ are larger than or equal to zero. Therefore by the definition of ${}^{c}D_{q}^{\beta }$, we get $({}^{c}D_{q}^{\beta } u) (t) \in C(\overline{J})$ and $({}^{c}D_{q}^{\beta } u) (t) \geq 0$. Now, define $\tau (t) = w_{n} ( t, f(t), f'(t), ({}^{c}D_{q}^{ \beta } f)(t) )$. Then $\tau \in L^{\kappa }(\overline{J})$ and $\tau (t)$ higher than or equal to m for almost all $t\in \overline{J}$. It follows from $G_{q} ( t , s) \geq 0$ for all $(t, s)$ belonging to $\overline{J} \times \overline{J}$, from the equality

$$\begin{aligned} (\varTheta _{n} u) (t) ={} & \frac{a \varGamma _{q} ( \alpha - \beta )( B- c_{2})(1 - qs)^{(\alpha -1)} }{( 1 -c_{1} ) \varGamma _{q} (\alpha - \beta ) ( \varGamma _{q} ( 2 - \beta ) - c_{2})} \bigl(I_{q}^{\alpha }\tau \bigr) (1) \\ & {} + \frac{ c_{2}B\varGamma _{q}(\alpha )( c_{1} + t - c_{1} t)(1 - qs)^{( \alpha -\beta -1)}}{( 1 - c_{1} ) \varGamma _{q} ( \alpha ) (B - c _{2})} \bigl(I_{q}^{\alpha -\beta } \tau \bigr) (1) + \bigl(I_{q}^{\alpha }\tau \bigr) (t). \end{aligned}$$

From the properties of $I_{q}^{\alpha }$ that $\varTheta _{n} u \in C( \overline{J})$ and $(\varTheta _{n} u)(t) \geq 0$ for all $t\in \overline{J}$ we have $(\varTheta _{n} u)'(t) = (I_{q}^{\alpha -1} \tau ) (t)$. Hence, $(\varTheta _{n} u)' \in C(\overline{J})$ and $(\varTheta _{n} u)'$ higher than or equal to zero, on J̅. We test that the operator $\varTheta _{n}$ is continuous. Suppose that the sequence ${u_{m}} \subset P$ is convergent and $\lim_{m \to \infty } u_{m} =u$. Thus, $\lim_{m\to \infty } u_{m}^{(i)} (t)= u^{(i)} (t)$ uniformly on J̅ for $i =0,1$. Since

$$ \bigl({}^{c}D_{q}^{\beta } u \bigr) (t) =\frac{d}{dt} \bigl(I_{q}^{1 - \beta }\bigr) \bigl( u(t) -u(0) \bigr) = \bigl(I_{q}^{1-\beta } u'\bigr) (t), $$

(7)

we get

$$\begin{aligned} \bigl| \bigl({}^{c}D_{q}^{\beta }u_{m} \bigr) (t) - \bigl({}^{c}D_{q}^{\beta }u\bigr) (t) \bigr| & \leq \frac{ \Vert u'_{m} - u' \Vert }{ \varGamma _{q}( 1-\beta )} \int _{0}^{t} (t - qs)^{( - \beta )} \, d_{q}s \leq \frac{ \Vert u_{m} - u \Vert _{\ast }}{ \varGamma _{q}(\beta )} \end{aligned}$$

and $\lim_{ m\to \infty } ({}^{c}D_{q}^{\beta }u_{m}) (t) = ({}^{c}D _{q}^{\beta }u) (t)$ uniformly on J̅. In addition, by using (7), we have $\vert ({}^{c}D_{q}^{\beta }u_{m}) (t) \vert \leq \frac{ u'_{m} }{ \varGamma _{q}( \beta )}$ and so

$$ \| \bigl({}^{c}D_{q}^{\beta }u_{m} \bigr) \| \leq \frac{ \Vert u'_{m} \Vert }{ \varGamma _{q}( \beta )}. $$

(8)

Put $\tau _{m} (t)= w_{n} (t, u_{m}(t), u'_{m}(t), ({}^{c}D_{q} ^{\beta }u_{m}) (t) )$ and $\tau (t) = w_{n} ( t , u(t), u'(t), ({}^{c}D_{q}^{\beta }u )(t) )$. Then $\lim_{m\to \infty } \tau _{m} (t) = \tau (t)$ and there exists $\mu \in L^{\kappa }( \overline{J})$ such that $0 \leq \tau _{m} (t) \leq \mu (t)$, for each t in J̅ and natural number m. Since $w_{n}$ is a $L^{\kappa }$-Carathéodory function, $\{ u_{m} \}$, $\{ ({}^{c}D _{q}^{\beta } u_{m})(t) \}$ are bounded in $C^{1}(\overline{J})$, $C(\overline{J})$, respectively. So, $\lim_{m \to \infty } ( \varTheta _{n} u_{m}) (t) = (\varTheta _{n} u) (t)$ uniformly on J̅. Since $\{\tau _{m}\}$ is $L^{\kappa }$-convergent on J̅, we conclude that $\lim_{m\to \infty } (\varTheta _{n} u_{m})' (t) = \lim_{m \to \infty } (I_{q}^{\alpha -1} \tau _{m}) (t) = (\varTheta _{n} u)' (t)$, uniformly on J̅. Hence, the operator $\varTheta _{n}$ is a continuous. We choose a positive constant r such that both $\| u_{m} \|$ and $\| u'_{m} \|$ are less than or equal to r for each natural number m, thus, we have $\varGamma _{q} (\beta ) \| ({}^{c}D _{q}^{\beta }u_{m}) (t) \| \leq r$ and

$$\begin{aligned} \biggl\vert \int _{0}^{t} (t-qs)^{(\alpha -2)} \tau _{m} (s) \, d_{q}s \biggr\vert & \leq \biggl( \int _{0}^{t} (t- qs)^{( (\alpha -2) p)} \, d _{q}s \biggr)^{ \frac{1}{ p}} \biggl( \int _{0}^{t} \bigl\vert \tau _{m}(s) \bigr\vert ^{ \kappa }\, d_{q}s \biggr)^{\frac{1}{\kappa }} \\ & \leq \biggl(\frac{ t^{d}}{d} \biggr)^{\frac{1}{p}} \Vert \tau _{m} \Vert _{\kappa }, \end{aligned}$$

(9)

for all m. On the other hand, the relations

$$\begin{aligned} 0 & \leq (\varTheta _{n} u_{m}) (t) = \int _{0}^{1} G_{q}(t, s) \tau _{m} (s) \, d_{q}s \leq \int _{0}^{1} G_{q}(t, s) \mu (s) \, d_{q}s \leq \frac{ \Vert \mu \Vert _{1}}{ \varGamma _{q} (\alpha )} \end{aligned}$$

and

$$\begin{aligned} 0 & \leq (\varTheta _{n} u_{m})' (t) = \bigl(I_{q}^{\alpha -1} \tau _{m}\bigr) (t) \leq \bigl(I_{q}^{\alpha -1} \mu \bigr) (t) \leq \frac{1}{ \varGamma _{q}(\alpha -1)} \biggl[ \frac{1}{ (\alpha -2)p + 1} \biggr]^{ \frac{1}{p}} \Vert \mu \Vert _{\kappa }, \end{aligned}$$

hold for each t and m and so $\{ \varTheta _{n} u_{m}\}$ is bounded in $C(\overline{J})$. Moreover, it follows from Lemma 4 that

$$\begin{aligned} \bigl\vert (\varTheta _{n} u_{m})' (t_{2}) - (\varTheta _{n} u _{m})' ( t_{1}) \bigr\vert &= \bigl\vert \bigl(I_{q}^{\alpha -1} \bigr) \bigl(\tau _{m} (t_{2}) - \tau _{m} (t_{1}) \bigr) \bigr\vert \\ & \leq \frac{ \Vert \tau _{m} \Vert _{\kappa }}{ \varGamma _{q}( \alpha -1)} \biggl[ \biggl( \frac{ t_{1}^{d} + ( t_{2} - t_{1})^{d} - t_{2}^{d}}{d} \biggr)^{ \frac{1}{p}} + \biggl( \frac{ (t_{2} -t_{1} )^{d}}{d} \biggr)^{ \frac{1}{p}} \biggr] \\ & \leq \frac{ \Vert \mu \Vert _{\kappa }}{ \varGamma _{q}( \alpha -1)} \biggl[ \biggl( \frac{t_{1}^{d} + (t_{2} - t_{1})^{d} - t_{2}^{d}}{d} \biggr) ^{\frac{1}{p}} + \biggl( \frac{( t_{2}- t_{1})^{d}}{d} \biggr)^{ \frac{1}{p}} \biggr], \end{aligned}$$

for each $t_{1}$ and $t_{2}$ belonging to J̅ such that $t_{1} \leq t_{2}$ is fulfilled. As a result, $\{ (\varTheta _{n} u_{m})' \}$ is equicontinuous on J̅. Consequently, based on the Arzelà–Ascoli theorem, $\{ \varTheta _{n} u_{m} \}$ is relatively compact in $C^{1}(\overline{J})$. Also, since $\varTheta _{n}$ is continuous, we conclude that the operator $\varTheta _{n}$ is completely continuous. □

Lemma 8

([61, 62])

LetXbe a Banach space, $P\subset X$a cone and $\mathcal{O}_{1}$and $\mathcal{O}_{2}$bounded open balls inXcentered at the origin with $\overline{\mathcal{O}}_{1} \subset \mathcal{O}_{2}$. A completely continuous operatorwmapping $P \cap ( \overline{\mathcal{O}}_{2} \backslash \mathcal{O}_{1})$intoPhas a fixed point whenever $\| w (u)\| \geq \| u \|$and $\|w (u)\| \leq \| u\|$for $u \in P \cap \partial \mathcal{O}_{1}$and $u \in P\cap \partial \mathcal{O}_{2}$, respectively.

Theorem 9

Letwsatisfy conditions (H1) and (H2). Then problem (1) has a solution $f_{n}$inPsuch that

$$ f_{n} \geq \frac{m \varLambda _{2} }{ \alpha -\beta }, \qquad f'_{n}(t) \geq \frac{m t^{\alpha -1} }{ \varGamma _{q}( \alpha )}, \quad \textit{and}\quad \bigl( {}^{c}D_{q}^{\beta } f_{n}\bigr) (t) \geq \frac{m t^{\alpha - \beta }}{ \varGamma _{q} (\alpha - \beta + 1)}, $$

(10)

for alltbelonging toJ̅and the natural numbern.

Proof

By using Lemma 7, one can conclude that the operator $\varTheta _{n} : P \to P$ is completely continuous. A function f is a solution of problem (1), whenever f solves the operator equation $f= \varTheta _{n} f$. Finally, we demonstrate $w_{n}$ in P is a fixed point of $\varTheta _{n}$ with desired continuousness. For this purpose, it is observed that

$$\begin{aligned} (\varTheta _{n} u ) (t) & = \int _{0}^{1} G_{q} (t, s ) w_{n} \bigl( s, u(s), u'(s), \bigl({}^{c}D_{q}^{\beta }u \bigr) (s) \bigr) \, d_{q}s \\ & \geq m \int _{0}^{1} G_{q}(t,s) \, d_{q}s \geq m \int _{0}^{1} (1-t)^{ \alpha }(1-qs)^{(\alpha -\beta -1)} \, d_{q}s \\ & = \frac{m \varLambda _{2}}{\alpha -\beta } \end{aligned}$$

(11)

and so $\|\varTheta _{n} u \|_{\ast }\geq \| \varTheta _{n} u \| \geq \frac{m \varLambda _{2}}{ \alpha -\beta }$. Put

$$ \mathcal{O}_{1} = \biggl\{ u \in X : \Vert u \Vert _{\ast }< \frac{m \varLambda _{2} }{\alpha -\beta } \biggr\} . $$

Then $\|\varTheta _{n} u\|_{\ast }\geq \| u\|_{\ast }$ for all u belonging to $P\cap \partial \mathcal{O}_{1}$. Let $v_{n} = g_{1} ( \frac{1}{n} ) + g_{2} ( \frac{1}{n} ) + g _{3} ( \frac{1}{n} )$. Inequality (7) implies that

$$\begin{aligned} \bigl\vert (\varTheta _{n} u) (t) \bigr\vert & \leq \biggl\vert \int _{0}^{1} G_{q}(t, s) w_{n} \bigl( s, f(s), f'(s), \bigl( {}^{c}D_{q}^{\beta }f\bigr) (s) \bigr) \, d_{q}s \biggr\vert \\ & \leq \int _{0}^{1} \bigl\vert G_{q}(t,s) \bigr\vert \bigl[v_{n} + \gamma (s) \theta \bigl(1+ \bigl\vert u(s) \bigr\vert , 1+ \bigl\vert u'(s) \bigr\vert , 1+ \bigl\vert \bigl({}^{c}D_{q}^{\beta }u\bigr) (s) \bigr\vert \bigr) \bigr] \, d_{q}s \\ & \leq \varLambda _{1} \bigl( v_{n} + w\bigl(1+ \Vert u \Vert , 1 + \bigl\Vert u' \bigr\Vert , 1+\bigl\| \bigl( {}^{c}D^{\beta }u\bigr) \bigr\Vert \bigr) \Vert \gamma \|_{1} \bigr) \end{aligned}$$

and

$$\begin{aligned} \bigl\vert (\varTheta _{n} u)'(t) \bigr\vert ={}& \bigl\vert \bigl(I_{q}^{\alpha -1} w _{n}\bigr) \bigl(t, u(t), u'(t), \bigl({}^{c}D_{q}^{\beta }u \bigr) (t) \bigr) \, d _{q}s \bigr\vert \\ \leq {}&\bigl(I_{q}^{\alpha -1} \bigl( v_{n} + \gamma (t) \theta \bigl(1+ \bigl\vert u(t) \bigr\vert , 1+ \bigl\vert u'(t) \bigr\vert , 1+ \bigl\vert \bigl({}^{c}D_{q}^{\beta }u \bigr) (t) \bigr\vert \bigr) \bigr) \bigr) \\ & {}+ \frac{v_{n} t^{\alpha -1}}{(\alpha -1) \varGamma _{q} (\alpha -1)} \\ & {} + w \bigl( 1 + \Vert u \Vert , 1+ \bigl\Vert u' \bigr\Vert , 1 + \bigl\Vert \bigl({}^{c}D_{q}^{ \beta }u \bigr) \bigr\Vert \bigr) \bigl(I_{q}^{\alpha -1} \gamma \bigr) (t), \end{aligned}$$

for each $u\in P$ and all $t\in \overline{J}$, because w is increasing in all its arguments. Since $\| u\|$ and $\|u'\|$ are less than or equal to $\|u\|_{\ast }$, $\| ({}^{c}D_{q}^{\beta }u) \| \leq \frac{ \| u' \| }{ \varGamma _{q} ( \beta )} \leq \frac{ \| u \|_{\ast }}{ \varGamma _{q} ( \beta )}$ and by inequality (9), $\int _{0}^{t} (t - qs)^{( \alpha -2)} \gamma (s) \, d_{q}s \leq (\frac{1}{d})^{1/p} \| \gamma \|_{\kappa }$, we have

$$ \bigl\Vert \varTheta _{n} (x) \bigr\Vert \leq \varLambda _{1} \biggl[ v_{n} + w \biggl(1 + \Vert u \Vert _{\ast }, 1+ \Vert u \Vert _{\ast }, 1 + \frac{ \Vert u \Vert _{\ast }}{ \varGamma _{q}( \beta )} \biggr) \Vert \gamma \Vert _{1} \biggr] $$

and

$$ \bigl\Vert (\varTheta _{n} u)' \bigr\Vert \leq \frac{1}{ \varGamma _{q}( \alpha -1)} \biggl[ \frac{v _{n}}{ \alpha -1} + w \biggl( 1+ \Vert u \Vert _{\ast }, 1 + \Vert u \Vert _{\ast }, 1 + \frac{ \Vert u \Vert _{\ast }}{ \varGamma _{q} (\beta )} \biggr) \biggl( \frac{1}{d} \biggr)^{1/p} \Vert \gamma \Vert _{\kappa } \biggr]. $$

Therefore,

$$ \Vert \varTheta _{n} u \Vert _{\ast }\leq M \biggl[ \frac{v_{n}}{ \alpha - 1} + N w \biggl( 1 + \Vert u \Vert _{\ast }, 1+ \Vert u \Vert _{\ast }, 1 + \frac{ \Vert u \Vert _{\ast }}{ \varGamma _{q} (\beta )} \biggr) \biggr], $$

where N and M are $\max \{ \| \gamma \|_{1}, ( \frac{1}{d} )^{1/p} \| \gamma \|_{\kappa } \}$ and $\max \{ \varLambda _{1}, \frac{1}{ \varGamma _{q} (\alpha -1)} \}$, respectively. Since

$$ \lim_{ v\to \infty } \frac{w(1+v ,1 + v, 1+ v)}{ v} $$

is equal to zero, by condition (H1), there exists a positive constant L such that

$$ M \biggl[ \frac{ v_{n}}{\alpha -1} + N w \biggl( 1+v, 1+v, \frac{v}{ \varGamma (\beta )} \biggr) \biggr] < v, $$

for each v higher than or equal to L. Hence, $\| \varTheta _{n} u\| _{\ast }< \| u\|_{\ast }$ for all u in P with $\| u \|_{\ast } \geq L$. Let $\mathcal{O}_{2} = \{ u \in X : \|u\|_{\ast }< L \}$, then $\|\theta _{n} u\|_{\ast }< \| u \|_{\ast }$ for $u \in P\cap \partial \varOmega _{2}$. Now applying the last result, with X and $w = \varTheta _{n}$, we conclude that $\varTheta _{n}$ has a fixed point $f_{n}$ in $P \cap (\overline{\mathcal{O}}_{2} \backslash \mathcal{O}_{1})$. Consequently, $f_{n}$ is a solution of Problem (1). The first inequality follows from (11), $f_{n} =( \varTheta _{n} f_{n})(t) \geq \frac{m \varLambda _{2} }{\alpha - \beta }$, the second one follows from the relation

$$ (\varTheta _{n} u)'(t) = \bigl(I_{q}^{\alpha -1} w_{n}\bigr) \bigl(t, u(t), u' (t), \bigl({}^{c}D_{q}^{\beta }u\bigr) (t) \bigr) \geq \bigl(I_{q}^{\alpha -1} m \bigr) = \frac{ m t^{\alpha -1}}{ \varGamma _{q}(\alpha )}, $$

for $t\in \overline{J}$ and u belongs to P. Finally, using the second inequality and $(I_{q}^{1- \beta } u)(t) = \frac{ \varGamma _{q} ( \alpha ) }{ \varGamma _{q} (\alpha -\beta +1)} t^{\alpha -\beta }$, where $u(t) = t^{\alpha -1}$, we obtain

$$ \bigl({}^{c}D_{q}^{\beta }f_{n} \bigr) (t) = \bigl(I_{q}^{1 - \beta } f'_{n} \bigr) (t) \geq \frac{m}{ \varGamma _{q}(\alpha ) } \bigl(I_{q}^{1-\beta } h \bigr) (t) = \frac{ m t^{\alpha -\beta }}{ \varGamma _{q} (\alpha - \beta +1)}, $$

for each t. This completes our proof. □

Theorem 10

The problem (1) has a solutionfsuch that $( \alpha - \beta ) f(t) \geq m \varLambda _{2}$, $\varGamma _{q}( \alpha ) f'(t) \geq m t^{\alpha -1}$and $\varGamma _{q}( \alpha -\beta +1) ({}^{c}D _{q}^{\beta }f) (t) \geq m t^{ \alpha -\beta }$, for all $t\in \overline{J}$, whenever conditions (H1) and (H2) hold.

Proof

By using Theorem 9, for each n, problem (1) has a solution $f_{n}\in P$ which satisfies inequality (10). Hence

$$ g_{1} \bigl(f_{n} (t)\bigr) \leq g_{1} \biggl( \frac{ m \varLambda _{2}}{ \alpha - \beta } \biggr), \qquad g_{2} \bigl( \bigl\vert f'_{n} (t) \bigr\vert \bigr) \leq g_{2} \biggl( \frac{ m t ^{ \alpha - 1}}{ \varGamma _{q} ( \alpha )} \biggr) $$

and

$$ g_{3} \bigl( \bigl\vert \bigl({}^{c}D_{q}^{ \beta } f_{n}\bigr) (t) \bigr\vert \bigr) \leq g _{3} \biggl( \frac{ m t^{ \alpha -\beta }}{ \varGamma _{q}( \alpha - \beta +1)} \biggr), $$

for each $t\in \overline{J}$ and all natural number n. In addition, it follows from (8) that $\| ({}^{c}D_{q}^{\beta }f_{n}) \| \leq \frac{ \| f'_{n} \|}{ \varGamma _{q} (\beta )}$. We put

$$ F(t) = g_{1} \biggl( \frac{m \varLambda _{2}}{ \alpha -\beta } \biggr) + g_{2} \biggl( \frac{m t^{\alpha -1}}{ \varGamma _{q}(\alpha )} \biggr) + g_{3} \biggl( \frac{ m t^{ \alpha -\beta }}{ \varGamma _{q}( \alpha - \beta +1)} \biggr). $$

(12)

Therefore, we conclude that

$$\begin{aligned} m & \leq w_{n} \bigl( t, f_{n}(t), f'_{n} (t), \bigl({}^{c}D_{q}^{\beta } f_{n}\bigr) (t) \bigr) \\ & \leq F(t) + \gamma (t) \theta \bigl( 1 + \Vert f_{n} \Vert , 1 + \bigl\Vert f'_{n} \bigr\Vert , 1 + \bigl\Vert \bigl({}^{c}D_{q}^{\beta }f_{n} \bigr) \bigr\Vert \bigr) \\ & \leq F(t) + \gamma (t) \theta \biggl( 1 + \Vert f_{n} \Vert _{\ast }, 1 + \bigl\Vert f'_{n} \bigr\Vert _{\ast }, 1 + \frac{ \Vert f_{n} \Vert _{\ast }}{ \varGamma _{q}( \beta )} \biggr). \end{aligned}$$

Since we have a positive value $G_{q}(t,s)\leq \varLambda _{1}$, we get

$$\begin{aligned} 0 &\leq f_{n} (t)= \int _{0}^{1} G_{q}(t,s) w_{n} \bigl(s, f_{n} (s), f'_{n} (s), \bigl({}^{c}D_{q}^{\beta }f_{n} \bigr) (s) \bigr) \, d_{q}s \\ & \leq \varLambda _{1} \biggl[ \int _{0}^{1} F(qs) \, d_{q}s + w \biggl( 1 + \Vert f_{n} \Vert _{\ast }, 1 + \Vert f_{n} \Vert _{\ast }, 1 + \frac{ \Vert f_{n} \Vert _{\ast }}{ \varGamma _{q} (\beta )} \biggr) \Vert \gamma \Vert _{1} \biggr] \end{aligned}$$

and

$$\begin{aligned} 0 &\leq f'_{n} (t) \leq \bigl(I_{q}^{\alpha -1} F\bigr) (t) + w \biggl(1 + \Vert f_{n} \Vert _{\ast }, 1 + \Vert f_{n} \Vert _{\ast }, 1 + \frac{ \Vert f_{n} \Vert _{\ast }}{ \varGamma _{q} (\beta )} \biggr) \bigl(I_{q} ^{\alpha -1} \gamma \bigr) (t). \end{aligned}$$

At present, we show that $\int _{0}^{t} (t- qs)^{(\alpha -2)} F(s)\, d _{q}s$ is bounded on $[0,1]$. By using the Hölder inequality, we get

$$\begin{aligned} &\int _{0}^{1}(t-qs)^{ (\alpha -2)} g_{1} \biggl( \frac{m \varLambda _{2} }{ \alpha -\beta } \biggr) \, d_{q}s \\ &\quad = g_{1} \biggl( \frac{m \varLambda _{2}}{ \alpha -\beta } \biggr) \int _{0}^{1} (t-qs)^{ (\alpha -2)} \, d_{q}s = \frac{1}{\alpha -1} g_{1} \biggl( \frac{m (1-t)^{\alpha }}{ \alpha -\beta } \biggr) =:\lambda _{1}, \\ &\int _{0}^{t} (t-qs)^{ (\alpha -2)} g_{2} \biggl( \frac{m s^{\alpha -1}}{ \varGamma _{q} (\alpha )} \biggr) \, d_{q}s \\ &\quad = \biggl( \frac{1}{d} \biggr)^{1/p} \biggl( \frac{ \varGamma _{q} ( \alpha )}{m} \biggr)^{ \frac{1}{ (\alpha -1) \kappa } } \biggl[ \int _{0}^{ ( \frac{m}{ \varGamma _{q}( \alpha )} )^{ \frac{1}{ \alpha -1}}} g_{2}^{\kappa } \bigl(s^{\alpha -1}\bigr) \, d_{q}s \biggr]^{1/ \kappa }=: \lambda _{2}, \end{aligned}$$

and analogously

$$\begin{aligned} &\int _{0}^{t} (t-qs)^{(\alpha -2)} g_{3} \biggl( \frac{m s^{\alpha - \beta }}{ \varGamma _{q} ( \alpha -\beta +1) } \biggr) \, d_{q}s \\ & \quad = \biggl( \frac{1}{d} \biggr)^{1/p} \biggl( \frac{ \varGamma _{q}( \alpha - \beta + 1)}{ m} \biggr)^{ \frac{1}{(\alpha - \beta )\kappa }} \biggl[ \int _{0}^{ (\frac{m}{ \varGamma _{q}( \alpha -\beta +1)})^{\frac{1}{ \alpha - \beta }} } g_{3}^{\kappa } \bigl(s ^{\alpha -\beta }\bigr) \, d_{q}s \biggr]^{1/\kappa } \\ &\quad =:\lambda _{3}. \end{aligned}$$

Note that (H1) guarantees $\lambda _{j} < \infty$ for $j=1, 2$ and 3. Hence, for all $t \in \overline{J}$, we obtain

$$ \int _{0}^{t} (t- qs)^{ (\alpha -2)} F(s)\, d_{q}s \leq \lambda , $$

where $\lambda =\lambda _{1} + \lambda _{2} + \lambda _{3}$. Also, we have

$$\begin{aligned} \int _{0}^{1} F(qs) \, d_{q}s \leq{} &\frac{1}{\alpha -1} g_{1} \biggl( \frac{m \varLambda _{2}}{ \alpha -\beta } \biggr) + \biggl( \frac{ \varGamma _{q}(\alpha )}{m} \biggr)^{ \frac{1}{ \alpha -1}} \int _{0}^{ ( \frac{ m}{ \varGamma _{q}(\alpha )} )^{\frac{1}{\alpha -1}}} g _{2} \bigl( s^{\alpha -1}\bigr) \, ds \\ & {} + \biggl(\frac{ \varGamma _{q} (\alpha - \beta + 1)}{ m} \biggr) ^{ \frac{1}{ \alpha -\beta }} \int _{0}^{ ( \frac{m}{ \varGamma _{q} (\alpha -\beta +1)} )^{ \frac{1}{ \alpha -\beta }}} g_{3} \bigl( s ^{ \alpha -\beta }\bigr) \, d_{q}s \\ < {}& \infty . \end{aligned}$$

Now, we conclude from the estimates

$$\begin{aligned} \Vert f_{n} \Vert & = \varLambda _{1} \biggl[ \int _{0}^{1} F(qs) \, d_{q}s + w \biggl( 1 + \Vert f_{n} \Vert _{\ast }, 1 + \Vert f_{n} \Vert _{\ast }, 1 + \frac{ \Vert f_{n} \Vert _{\ast }}{\varGamma _{q}( \beta )} \biggr) \Vert \gamma \Vert _{1} \biggr] \end{aligned}$$

and

$$\begin{aligned} \bigl\Vert f'_{n} \bigr\Vert & \leq \frac{1}{\varGamma _{q}(\alpha - 1)} \biggl[ \lambda + w \biggl(1 + \Vert f_{n} \Vert _{\ast }, 1 + \Vert f_{n} \Vert _{\ast }, 1+ \frac{ \Vert f _{n} \Vert _{\ast }}{ \varGamma _{q}( \beta )} \biggr) \biggl( \frac{1}{d} \biggr) ^{1/p} \Vert \gamma \Vert _{\kappa } \biggr] \end{aligned}$$

to the inequality

$$ \Vert f_{n} \Vert _{\ast }\leq M \biggl[ \eta _{1} + \eta _{2} w \biggl( 1 + \Vert f_{n} \Vert _{\ast }, 1 + \Vert f_{n} \Vert _{\ast }, 1 + \frac{ \Vert f_{n} \Vert _{\ast }}{ \varGamma _{q}(\beta )} \biggr) \biggr], $$

(13)

holding, for $n \geq 1$, where $M=\max \{ \varLambda _{1}, \frac{1}{ \varGamma _{q}( \alpha - 1)} \}$, $\eta _{1} = \max \{ \lambda , \int _{0}^{1} F(qs) \, d_{q}s \}$ and

$$ \eta _{2} = \max \biggl\{ \Vert \gamma \Vert _{1}, \biggl( \frac{1}{d} \biggr) ^{ 1/p} \Vert \gamma \Vert _{\kappa } \biggr\} . $$

Now, by condition (H1), there exists a positive constant L such that

$$ M \biggl[ \eta _{1} + \eta _{2} w \biggl( 1 + v, 1+ v, 1 + \frac{v}{ \varGamma _{q}( \beta )} \biggr) \biggr] < v, $$

for each v higher than or equal to L. Now, inequality (13) gives $\| f_{n}\|_{\ast }< L$, for all n. Therefore

$$ w_{n} \bigl(t, f_{n} (t), f'_{n} (t), \bigl({}^{c}D_{q}^{\beta }f_{n} \bigr) (t) \bigr) \leq R(t), $$

where $R(t) = F(t) + \gamma (t) \theta ( 1 + L, 1 + L, 1 + \frac{L}{ \varGamma _{q}( \beta )} )$. Note that, from condition (H1), R in $L^{\kappa }(\overline{J})$. Let

$$ \tau _{n} (t) = w_{n} \bigl(t, f_{n} (t),f'_{n} (t), \bigl({}^{c}D_{q}^{ \beta } f_{n}\bigr) (t) \bigr) $$

and $t_{1} , t_{2} \in [0, \delta ]$ such that $t_{1} \leq t_{2}$. Then

$$\begin{aligned} \bigl\vert f'_{n} (t_{2} ) - f'_{n} (t_{1} ) \bigr\vert ={}& \bigl(I_{q}^{\alpha -1}\bigr) \bigl\vert \tau _{n} (t_{2})- \tau _{n} (t_{1}) \bigr\vert \\ \leq{}& \frac{1}{ \varGamma _{q}(\alpha -1)} \biggl[ \int _{0}^{ t_{1}} \bigl( (t_{1} -qs)^{(\alpha -2)} - ( t_{2} - qs)^{(\alpha -2)} \bigr) \tau _{n} (s) \, d_{q}s \\ & {} + \int _{t_{1}}^{t_{2}} (t_{2} -qs)^{(\alpha -2)} \tau _{n} (s) \, d_{q}s \biggr] \\ \leq{}& \frac{1}{ \varGamma _{q}(\alpha -1)} \biggl[ \int _{0}^{t_{1}} \bigl( (t_{1} -qs)^{(\alpha -2)} - (t_{2} -qs)^{(\alpha -2)} \bigr) R(s) \, d_{q}s \\ & {} + \int _{t_{1}}^{t_{2}} (t_{2} -qs )^{(\alpha -2)} R(s)\, d _{q}s \biggr] \end{aligned}$$

and so, by applying Lemma 4, we get

$$ \bigl\vert f'_{n} (t_{2}) - f'_{n} (t_{1}) \bigr\vert \leq \frac{ \Vert R \Vert _{\kappa }}{ \varGamma _{q}(\alpha -1)} \biggl[ \biggl( \frac{t_{1}^{d} + (t_{2} -t_{1} )^{d} - t_{2}^{d}}{d} \biggr)^{1/p} + \biggl( \frac{(t _{2} - t_{1})^{d}}{d} \biggr)^{1/p} \biggr]. $$

As a consequence, $\{f'_{n}\}$ is equicontinuous on J̅. Since $\{f_{n}\}$ is bounded in $C(\overline{J})$, without less of generality, we may assume that $\{f_{n}\}$ is convergent in $C(\overline{J})$ by the Arzelà–Ascoli theorem. Let $\lim_{n \to \infty } f_{n} = f$, then passing to the limit as $n\to \infty$, we obtain $({}^{c}D_{q}^{\beta }f_{n}) (t) = (I_{q} ^{\alpha -1} f'_{n}) (t)$ and using Eq. (7), we have

$$ \lim_{n \to \infty } \bigl({}^{c}D_{q}^{\beta } f_{n}\bigr) (t) = \frac{1}{ \varGamma _{q}( \alpha -1)} \int _{0}^{t} (t - qs)^{(-\beta )} f' (s) \, d _{q}s, $$

uniformly on J̅. The last relation yields $\lim_{n\to \infty } ({}^{c}D_{q}^{\beta }f_{n}) (t) = ({}^{c} D_{q} ^{\beta }f) (t)$ in $C(\overline{J})$. Hence,

$$ \lim_{n \to \infty } w_{n} \bigl(t, f_{n} (t), f'_{n} (t), \bigl({}^{c}D _{q}^{\beta }f_{n}\bigr) (t) \bigr) = w \bigl( t, f(t), f'(t), \bigl({}^{c}D_{q} ^{\beta }f\bigr) (t) \bigr). $$

Since $R\in L^{\kappa }(\overline{J})$, by taking $n\to \infty$ in the equality

$$ f_{n} (t) = \int _{0}^{1} G_{q}(t,s) w_{n} \bigl(s, f_{n} (s), f'_{n} (s), \bigl({}^{c}D_{q}^{\beta }f_{n} \bigr) (s) \bigr) \, d_{q}s. $$

By using the dominated convergence theorem for $L^{\kappa }( \overline{J})$, we get

$$ f (t) = \int _{0}^{1} G_{q}(t,s) w \bigl( s, f (s), f'(s), \bigl({}^{c}D_{q} ^{\beta }f\bigr) (s) \bigr) \, d_{q}s. $$

Consequently, f is a solution of problem (1), satisfying the boundary conditions. This completes our proof. □

4 Algorithms and examples

In this section, we give some algorithms to illustrate problem (1), in Theorems 10 and present numerical examples. Foremost, we present a simplified analysis that can be executed to calculate the value of q-Gamma function, $\varGamma _{q} (x)$, for input q, x and different values of n. To this aim, we consider a pseudo-code description of the method for calculating the q-Gamma function of order n in Algorithm 2 (for details, see the link https://en.wikipedia.org/wiki/Q-gamma_function). Now we give some examples to illustrate our results. Table 1 shows that when q is constant, the q-Gamma function is an increasing function. Also, for smaller values of x, an approximate result is obtained with smaller values of n. It is shown by underlined rows. Table 2 shows that the q-Gamma function for values q close to 1 is obtained with higher values of n in comparison with other columns. They have been underlined in line 8 of the first column, line 17 of the second column and line 29 of third columns of Table 2. Also, Table 3 is the same as Table 2, but x values increased in 3. Similarly, the q-Gamma function for values of q close to 1 is obtained with higher values of n in comparison with other columns.

Table 1 Some numerical results for calculation of $\varGamma _{q}(x)$ with $q=\frac{1}{3}$ that is constant, $x=4.5, 8.4, 12.7$ and $n=1, 2, \ldots , 15$ of Algorithm 2

Table 2 Some numerical results for calculation of $\varGamma _{q}(x)$ with $q=\frac{1}{3}, \frac{1}{2}, \frac{2}{3}$, $x=5$ and $n=1, 2, \ldots , 35$ of Algorithm 2

Table 3 Some numerical results for calculation of $\varGamma _{q}(x)$ with $x=8.4$, $q=\frac{1}{3}, \frac{1}{2}, \frac{2}{3}$ and $n=1, 2, \ldots , 40$ of Algorithm 2

Here, we provide an example to illustrate our main result.

Example 1

Let $\overline{J}=[0,1]$, $\tau _{1}$ and $\tau _{2}$ belongs to $L^{\kappa }(\overline{J})$ and $\tau _{1} (t)$ higher than or equal to positive real number m for all $t\in \overline{J}$. Also, let

$$\begin{aligned} w(t, x_{1}, x_{2}, x_{3}) =& \tau _{1} (t) + \frac{1}{ x_{1}^{2/5} - r} + \frac{1}{ x_{2}^{1/4}} + \frac{1}{ x_{3}^{1/4}} \\ &{}+ \bigl\vert \tau _{2} (t) \bigr\vert \bigl( x_{1}^{ 2/5} + x_{2}^{1/4} + x_{3}^{1/4} \bigr), \end{aligned}$$

on $\overline{J}\times A$ with $A=[0, \infty ) \times [0, \infty ) \times [0, \infty )$, $g_{1}( u) = \frac{1}{ u^{2/5} - r}$ whenever $u^{2/5} \geq r$ and $g_{1}(u) = 0$ whenever $u^{2/5} < r$, $g_{2}(u) = \frac{1}{u^{1/4}}$, $g_{3}(u) = \frac{1}{u^{1/4}}$,

$$ w(x_{1}, x_{2} , x_{3}) = x_{1}^{2/5} + x_{2}^{1/4} + x_{3}^{1/4} + 1 $$

and $\gamma (t) = \tau _{1} (t) + |\tau _{2}(t)|$, where $r =( a f(1) )^{2/5}$. Since w satisfies conditions (H1) and (H2), Theorem 10 guarantees that problem (1) has a positive solution.

Example 2

In this example, we choose a problem similar to (1),

$$ \textstyle\begin{cases} {}^{c}D_{q}^{9/4} f(t) = t + 1 + \frac{1}{ (f (t))^{2/5} - \lambda } + \frac{1}{ (f'(t))^{1/4}} + \frac{1}{ [ ({}^{c}D_{q}^{1/4} f )(t) ]^{1/4}} \\ \hphantom{{}^{c}D_{q}^{9/4} f(t) = } {}+ 2 ( f(t)^{2/5} + f'(t)^{1/4} + [ ( {}^{c}D_{q} ^{1/4} f )(t) ]^{1/4} + 1 ), \\ f(0) = \frac{1}{4} f(1), \\ f'(0) = \frac{1}{3} ( {}^{c}D_{q}^{1/4} f )(1), \\ f''(0) = f'''(0)= \cdots = f^{(n-1)}(0)=0, \end{cases} $$

where $\lambda = (\frac{1}{4} f(1))^{1/3}$. here $\alpha =\frac{9}{4} \in (2, 3)$, with $n=3$, $\beta =\frac{1}{4} \in (0,1)$, $c_{1}= \frac{1}{4} \in (0,1)$, $c_{2}= \frac{1}{3} \in (0, \varGamma _{q} ( \frac{7}{4}))$ for all $q \in (0, 1)$ and $\kappa (\frac{9}{4} -1) = \frac{4}{5} > 1$. Then

$$ \begin{aligned} w \bigl( t, f(t), f'(t), \bigl( {}^{c}D_{q}^{1/4} f \bigr) (t) \bigr) ={}& t + 1 + \frac{1}{ f (t)^{1/3} - \lambda } + \frac{1}{ f'(t)^{1/4}} \\ & {} + \frac{1}{ [ ({}^{c}D_{q}^{1/4} f )(t) ] ^{1/4}} \\ & {} + 2 \bigl(f (t)^{1/3} + r'(t) ^{1/4} + \bigl[ {}^{c}D_{q} ^{1/4} f(t) \bigr]^{1/4} +1 \bigr), \end{aligned} $$

and w may be singular at $t=0$ and satisfies conditions (H1) and (H2), for $g_{1}( h) = \frac{1}{ h^{1/3}- \lambda }$ whenever $h^{1/3}- k \geq 0$ and $g_{1}( h) = 0$ whenever $h^{1/3}- \lambda < 0$, $g_{2}(h)= \frac{1}{h^{1/4}}$, $g_{3}(h)= \frac{1}{h^{1/4}}$,

$$ w(x_{1}, x_{2}, x_{3}) =x_{1}^{1/3} + x_{2}^{1/4} + x_{3}^{1/4} + 1 $$

and $\tau _{1} (t)= t + 1 > 1 = m$, $\tau _{2} (t) = 2$ and $\gamma (t) = \tau _{1} (t) + |\tau _{2} (t)|$, Theorem 10 guarantees that problem (1) has a positive solution. Now, we investigate the computational complexity of Example 2 of Algorithm 6 and 7. Note that n in Algorithms 6 and 7 is used for calculating $\varGamma _{q}(x)$. Tables 4, 5 and 6 show the values of $\varLambda _{1}$ and $\varLambda _{2}$ for $q=\frac{1}{3}, \frac{1}{2}$ and $\frac{3}{4}$, respectively, an approximate result is obtained with less than four decimal places indicated by underlining.

Algorithm 6

The proposed method for calculating $\varLambda _{1}$

Algorithm 7

The proposed method for calculating $\varLambda _{2}$

Table 4 Some numerical results for calculation of $\varLambda _{1}$ and $\varLambda _{2}$ with $q=\frac{1}{3}$ and $n=1, 2, \ldots , 12$ of Example 2

Table 5 Some numerical results for calculation of $\varLambda _{1}$ and $\varLambda _{2}$ with $q=\frac{1}{2}$ and $n=1, 2, \ldots , 19$ of Example 2

Table 6 Some numerical results for calculation of $\varLambda _{1}$ and $\varLambda _{2}$ with $q=\frac{3}{4}$ and $n=1, 2, \ldots , 30$ of Example 2